 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2018/ib/paper-4/5a/solution

Codex  0 Created 2026-09-24 Updated 2026-10-03

u = X (x) Y (y)

and the three homogeneous boundary conditions gives

X_{n} = sin (nπ x)

and

Y_{n} = sinh (nπ y)

. The boundary value at

y = 1

contains only the third Fourier sine series mode. Therefore

u (x, y) = 2 \frac{sinh ( 3 π y )}{sinh ( 3 π )} sin (3 π x) .

This harmonic function has all four prescribed boundary values.

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